Bias-Optimality

We will consider a very broad class of problems.
A problem is defined by a
recursive procedure that takes as an input any potential solution
(a finite symbol string , where represents
a search space of solution candidates)
and outputs 1 if is a solution to ,
and 0 otherwise. Typically the goal is to find
as quickly as possible some that solves .

Define a probability distribution on a finite or infinite
set of programs for a given computer. represents the searcher's initial bias
(e.g., could be based on program length, or on a probabilistic syntax diagram).
A bias-optimal searcher will not spend more
time on any solution candidate than it deserves,
namely, not more than the candidate's probability times the total search time:

Definition 2.1 (BIAS-OPTIMAL SEARCHERS)
Let be a problem class,
a search space of solution candidates
(where any problem should have a solution in ),
a task-dependent bias in the form of conditional probability
distributions on the candidates . Suppose that we also have
a predefined procedure that creates and tests any given
on any within time (typically unknown in advance).
A searcher is -bias-optimal () if
for any maximal total search time
it is guaranteed to solve any problem
if it has a solution
satisfying
.
It is bias-optimal if .

This definition makes intuitive sense: the most probable candidates
should get the lion's share of the total search time, in a way that
precisely reflects the initial bias.
Still, bias-optimality is a particular restricted notion of optimality,
and there may be situations where it is not the most appropriate
one [59]--compare section 5.4.